Question

consider the polynomial expression \(5 x+6-3 x^{3}-4 x^{2}\) Write the expression in standard form and name its terms.

Short answer

The polynomial \(5 x+6-3 x^{3}-4 x^{2}\) in standard form is \(-3 x^{3}-4 x^{2}+5 x+6\). The terms are \(-3x^3\), \(-4x^2\), \(5x\), and \(6\).

Step-by-step solution

Order the terms

Rearrange the given expression into standard form where terms are ordered from highest to lowest exponent on the variable. Hence, we have \(-3 x^{3}-4 x^{2}+5 x+6\).

Identify the Terms

Identify the terms of the polynomial. Each term is a monomial that is separated by a \('+\' or \('-'\) sign, and contributes uniquely to the value of the polynomial. Here, \(-3x^3\), \(-4x^2\), \(5x\), and \(6\) are the terms.

Key concepts

Polynomial Expressions

Polynomial expressions are algebraic expressions that consist of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. These expressions represent a sum of terms, where each term is a product of a number—called the coefficient—and a whole number power of variables.

For instance, in the polynomial expression \(5x + 6 - 3x^3 - 4x^2\), the variables \(x\), \(x^2\), and \(x^3\) are raised to powers known as exponents. The numbers before them, namely 5, 6, -3, and -4, are the coefficients. This particular example is a cubic polynomial because the highest power of the variable \(x\) is 3. Understanding how to interpret and work with these expressions is a fundamental skill in algebra that enables students to solve a wide range of mathematical problems.

Ordering Polynomial Terms

Ordering the terms in a polynomial expression is critical for simplifying calculations and understanding the structure of the polynomial. The standard form of a polynomial arranges the terms in descending order by degree, which refers to the highest power of the variable in each term.

The process involves looking at each term's exponent and placing the terms from highest to lowest exponent. For example, the polynomial \(5x + 6 - 3x^3 - 4x^2\) can be rearranged to \( -3x^3 - 4x^2 + 5x + 6\), where the terms are ordered by the degrees 3, 2, 1, and 0, respectively. This is the standard form. Writing polynomials in standard form is vital for clarity and is especially useful when adding, subtracting, or comparing polynomials.

Identifying Polynomial Terms

Identifying individual terms in a polynomial paves the way for operations such as addition, subtraction, and understanding the functionality of the polynomial. Each term, also known as a monomial, is a single algebraic expression that stands alone, separated from other terms by a plus \(+\) or minus \( -\) sign. It includes a coefficient and possibly a variable raised to an exponent.

In the expression \( -3x^3 - 4x^2 + 5x + 6\), there are four terms: \( -3x^3\), \( -4x^2\), \(5x\), and \(6\). Notably, the constant term \(6\) has an implied exponent of 0, since any variable raised to the zero power is 1. Recognizing each term allows for a better grasp of the polynomial’s behavior, and is a necessary step before performing operations such as factoring or deriving.